This solution is from Arun Iyer of SIA High School and Junior College.

Part 1 First I will show that POR is a straight line. For this I would like to state the perpendicular bisector theorem.

PERPENDICULAR BISECTOR THEOREM: Every point equidistant from the two ends of a line segment lies on the perpendicular bisector of the line segment.

Now consider the line segment QS.

OQ=OS=1 therefore by the perpendicular bisector theorem, O must lie on the perpendicular bisector of QS.

PQ=PS (as the sides of the rhombus are equal), therefore by the perpendicular bisector theorem, P must lie on the perpendicular bisector of QS

RQ=RS (as the sides of the rhombus are equal), therefore by the perpendicular bisector theorem, R must lie on the perpendicular bisector of QS.

Now the perpendicular bisector of a line segment is unique and hence P, O, R must lie on the same perpendicular bisector and hence POR is a straight line.

Part 2 Now I will get all the angles of the rhombus.

ÐQPS = 72^o (given), ÐQRS = ÐQPS = 72^o as they are opposite angles of a rhombus. The diagonal of the rhombus bisects the angles of a rhombus and therefore ÐQPO = ÐSPO = ÐQRO = ÐSRO = 36^o.

Triangles OQR and OSR are isosceles triangles, therefore ÐOSR = ÐOQR = 36^o.

Using the fact that sum of angles of a triangle is 180 degrees, we can see that ÐSOR and ÐQOR are equal to 108 degrees. Since we have proved that POR is a straight line in Part 1, we can determine ÐQOP and ÐSOP to be 72 degrees.

Again using the fact that sum of angles of a triangle is 180 degrees, we can see that ÐOQP and ÐOSP are equal to 72 degrees.

Part 3 Let the side of the rhombus be x.

Consider triangle OPS in which PO=PS=x (since ÐPOS = ÐPSO).

Applying the cosine rule

cosOPS = [PS2 + PO2 - OS2]/[2× PS ×PO]

therefore

cos36 = [2x2-1]/[2x2]    (1).

Splitting the isosceles triangle ORS into two right angled triangles gives

cos36 = x/2     (2).

From (1) and (2),

[2x2 - 1]/[2x2] = x/2 x3 - 2x2 + 1 = 0 (x - 1)(x2 - x - 1) = 0.

Now x ¹ 1 because triangle POS is not equilateral, therefore x² - x - 1 = 0 and hence

x = [1 + Ö5]/2 or x = [1 - Ö5]/2.

Clearly x ¹ [1 - Ö5]/2 because the side length cannot be negative, therefore x = [1 + Ö5]/2, the Golden Ratio.